Quantum computation is conventionally performed using quantum operations acting on two-level quantum bits, or qubits. Qubits in modern quantum computers suffer from inevitable detrimental interactions with the environment that cause errors during computation, with multi-qubit operations often being a primary limitation. Most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. Qudits offer a larger state space to store and process quantum information, reducing complexity of quantum circuits and improving efficiency of quantum algorithms. Here, we experimentally demonstrate a ternary decomposition of a multi-qubit operation on cloud-enabled fixed-frequency superconducting transmons. Specifically, we realize an order-preserving Toffoli gate consisting of four two-transmon operations, whereas the optimal order-preserving binary decomposition uses eight \texttt{CNOT}s on a linear transmon topology. Both decompositions are benchmarked via truth table fidelity where the ternary approach outperforms on most sets of transmons on \texttt{ibmq\_jakarta}, and is further benchmarked via quantum process tomography on one set of transmons to achieve an average gate fidelity of 78.00\% $\pm$ 1.93\%.
Quantum computation is conventionally performed using quantum operations acting on two-level quantum bits, or qubits. Qubits in modern quantum computers suffer from inevitable detrimental interactions with the environment that cause errors during computation, with multi-qubit operations often being a primary limitation. Most quantum devices naturally have multiple accessible energy levels beyond the lowest two traditionally used to define a qubit. Qudits offer a larger state space to store and process quantum information, reducing complexity of quantum circuits and improving efficiency of quantum algorithms. Here, we experimentally demonstrate a ternary decomposition of a multi-qubit operation on cloud-enabled fixed-frequency superconducting transmons. Specifically, we realize an order-preserving Toffoli gate consisting of four two-transmon operations, whereas the optimal order-preserving binary decomposition uses eight \texttt{CNOT}s on a linear transmon topology. Both decompositions are benchmarked via truth table fidelity where the ternary approach outperforms on most sets of transmons on \texttt{ibmq_jakarta}, and is further benchmarked via quantum process tomography on one set of transmons to achieve an average gate fidelity of 78.00% ± 1.93%.
Entangling gates in superconducting quantum computers suffer from imperfection of control and relaxation of qubits. The Toffoli gate, essential to various quantum algorithms and to certain fast error correction schemes, represents a particular challenge, as it is decomposed into at least six two-qubit gates, whose errors accrue on top of the increased probability of qubit relaxation due to the long gate time. While it is possible to significantly reduce the depth of the Toffoli gate by exploiting three-level, i.e., qutrit states internally, existing methods are susceptible to errors arising from frequency drifts due to charge fluctuation effects in the second excited state of the transmon. In this poster, we propose a novel qutrit-based implementation of the Toffoli gate that is inherently robust against such errors. The method is extensively validated through a pulse-level simulation and experiments performed on IBM Quantum machines. On one device, where the total Toffoli gate time is 2.510μs, we obtained average gate fidelity values of 0.928 ± 0.007 and 0.896 ± 0.036 from multiple measurements performed during the first one and twenty-four hours after calibrations, respectively, where the errors represent the standard deviations of the measurement results.
Abstract Ternary quantum information processing in superconducting devices poses a promising alternative to its more popular binary counterpart through larger, more connected computational spaces and proposed advantages in quantum simulation and error correction. Although generally operated as qubits, transmons have readily addressable higher levels, making them natural candidates for operation as quantum three-level systems (qutrits). Recent works in transmon devices have realized high fidelity single qutrit operation. Nonetheless, effectively engineering a high-fidelity two-qutrit entanglement remains a central challenge for realizing qutrit processing in a transmon device. In this work, we apply the differential AC Stark shift to implement a flexible, microwave-activated, and dynamic cross-Kerr entanglement between two fixed-frequency transmon qutrits, expanding on work performed for the Z Z interaction with transmon qubits. We then use this interaction to engineer efficient, high-fidelity qutrit CZ † and CZ gates, with estimated process fidelities of 97.3(1)% and 95.2(3)% respectively, a significant step forward for operating qutrits on a multi-transmon device.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qutrits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic depth (runtime) decomposition of the Generalized Toffoli gate using no ancilla-a significant improvement over linear depth for the best qubit-only equivalent. Our circuit construction also features a 70x improvement in two-qudit gate count over the qubit-only equivalent decomposition. This results in circuit cost reductions for important algorithms like quantum neurons and Grover search. We develop an open-source circuit simulator for qutrits, along with realistic near-term noise models which account for the cost of operating qutrits. Simulation results for these noise models indicate over 90% mean reliability (fidelity) for our circuit construction, versus under 30% for the qubit-only baseline. These results suggest that qutrits offer a promising path towards scaling quantum computation.
The fidelity of certain gates on noisy quantum computers may be improved when they are implemented using more than two levels of the involved transmons. The main impediments to achieving this potential are the dynamic gate phase errors that cannot be corrected via calibration. The standard tool for countering such phase errors in two-level qubits is the echo protocol, often referred to as the dynamical decoupling sequence, where the evolution of a qubit is punctuated by an even number of X gates. We introduce basis cycling, which is a direct generalization of the qubit echo to general qudits, and provide an analytic framework for designing gate sequences to produce desired effects using this technique. We then apply basis cycling to a Toffoli gate decomposition incorporating a qutrit and obtain CCZ gate fidelity values up to 93.8$\pm$0.1%, measured by quantum process tomography, on IBM quantum computers. The gate fidelity remains stable without recalibration even while the resonant frequency of the qutrit fluctuates, highlighting the dynamical nature of phase error cancellation through basis cycling. Our results demonstrate that one of the biggest difficulties in implementing qudit-based gate decompositions on superconducting quantum computers can be systematically overcome when certain conditions are met, and thus open a path toward fulfilling the promise of qudits as circuit optimization agents.
Efficient Synthesis of Multi-Controlled Toffoli Gates with Ternary Clifford+P9 Gates
Temporary occupation of qutrit levels can reduce the width and depth required for binary circuit logic in quantum computing. We introduce an efficient intermediate-qutrit decomposition of multi-controlled Toffoli gates with binary-subspace inputs and outputs. For balanced control widths n=2h−1, the proposed decomposition is evaluated hierarchically through a tree, allowing independent subtree computations to proceed in parallel and reducing the depth from linear to logarithmic. The balanced low-depth construction requires 6n+3 logical P9 injections and 4n−3 clean ancillary qutrits. It matches the direct P9 count of a recursively extended Clifford+P9 baseline derived from the state-of-the-art work while using asymptotically one-quarter as many clean ancillas and replacing linear depth with logarithmic depth. For arbitrary control widths, a sequential extension preserves the 6n+3P9 count but has depth O(logm+n−m) for a balanced core m=2h−1≤n, which is linear in the worst case over n. By reducing the resource overhead of a widely used reversible primitive, the proposed decomposition provides a practical building block for the design and compilation of more resource-efficient quantum algorithms in the fault-tolerant regime.